• Li, Xiang (College of Science National University of Defense Technology) ;
  • Liu, Yicheng (College of Science National University of Defense Technology) ;
  • Wu, Jun (College of Mathematics and Computer Science Changsha University of Science Technology)
  • Received : 2015.08.06
  • Published : 2016.09.30


Self-organizing systems arise very naturally in artificial intelligence, and in physical, biological and social sciences. In this paper, we modify the classic Cucker-Smale model at both microscopic and macroscopic levels by taking the target motion pattern driving forces into consideration. Such target motion pattern driving force functions are properly defined for the line-shaped motion pattern and the ball-shaped motion pattern. For the modified Cucker-Smale model with the prescribed line-shaped motion pattern, we have analytically shown that there is a flocking pattern with an asymptotic flocking velocity. This is illustrated by numerical simulations using both symmetric and non-symmetric pairwise influence functions. For the modified Cucker-Smale model with the prescribed ball-shaped motion pattern, our simulations suggest that the solution also converges to the prescribed motion pattern.


  1. G. Albi and L. Pareschi, Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics, Appl. Math. Lett. 26 (2013), no. 4, 397-401.
  2. A. Barbaro, K. Taylor, P. F. Trethewey, L. Youse, and B. Birnir, Discrete and continuous models of the dynamics of pelagics fish: application to the capelin, Math. Comput. Simulation 79 (2009), no. 12, 3397-3414.
  3. B. Birnir, An ODE model of the motion of pelagics fish, J. Stat. Phys. 128 (2007), no. 1-2, 535-568.
  4. J. A. Canizo, J. A. Carrillo, and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci. 21 (2011), no. 3, 515-539.
  5. J. A. Carrillo, M. Fornasier, J. Rosado, and G.Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. 42 (2010), no. 1, 218-236.
  6. Y. L. Chuang, M. R. DOrsogna, D. Marthaler, A. L. Bertozzi, and L. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phy. D 232 (2007), no. 1, 33-47.
  7. I. D. Couzin, J. Krause, N. R. Franks, and S. Levin, Eective leadership and decision making in animal groups on the move, Nature 433 (2005), 513-516.
  8. F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math. 2 (2007), no. 1, 197-227.
  9. F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852-862.
  10. M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes, Slf-propelled particles with soft-core interactions: patterns, stability, and collapse, Phys. Rev. Lett. 96 (2006), 104-302.
  11. S. Y. Ha and J. G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297-325.
  12. S. Y. Ha and E. Tadmor, From particle to Kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008), no. 3, 415-435.
  13. Y. C. Liu and J. H. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl. 415 (2014), no. 1, 53-61.
  14. S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 144 (2011), no. 5, 923-947.
  15. J. Park, H. J. Kim, and S. Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control 55 (2010), no. 11, 2617-2623.
  16. J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math. 68 (2008), no. 3, 694-719.
  17. C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol. 68 (2006), no. 7, 1601-1623.